128 THE SQUARING OF THE CIRCLE. 



times the value 3^ for n and sometimes even the rougher ap- 

 proximation ?r = 3. The astronomer Ptolemy, who lived in Alex- 

 andria about the year 150 A. D., and who was famous as being the 

 author of the planetary system universally recognised as correct 

 down to the time of Copernicus, was the only one who furnished a 

 more exact value ; this he designated, in the sexagesimal system of 

 fractional notation which he employed, by 3, 8, 30, that is 3 and 

 -fa and ^f %-Q, or as we now say 3 degrees, 8 minutes (partes minutae 

 primae), and 30 seconds (paries minutae secundae}. As a matter of 

 fact, the expression 3 -j- -fa -f ^f^ = 3^ represents the number n 

 more exactly than 3^ ; but on the other hand, is, by reason of the 

 magnitude of the numbers 17 and 120 as compared with the num- 

 bers i and 7, more cumbersome. 



IV. 



THE ROMANS, HINDUS, CHINESE, ARABS, AND THE. CHRISTIAN 

 NATIONS TO THE TIME OF NEWTON. 



In the mathematical sciences, more than in any other, the Ro- 

 mans stood upon the shoulders of the Greeks. Indeed, with re- 

 spect to cyclometry, they not only did not add anything new to the 

 Grecian discoveries, but frequently even evinced that they either 

 did not know of the beautiful result obtained by Archimedes, or at 

 least could not appreciate it. For instance, Vitruvius, who lived 

 during the time of Augustus, computed that a wheel 4 feet in diam- 

 eter must measure 12 J feet in circumference; in other words, he 

 made n equal to 3^. And, similarly, a treatise on surveying, pre- 

 served to us in the Gudian manuscript of the library of Wolfen- 

 buttel, contains the following instructions for squaring the circle : 

 Divide the circumference of a circle into four parts and make one 

 part the side of a square ; this square will be equal in area to the 

 circle. Apart from the fact that the rectification of the arc of a 

 circle is requisite to the construction of a square of this kind, the 

 Roman quadrature, viewed as a calculation, is more inexact even 

 than any other known computation ; for its result is that ?r = 4. 



The mathematical performances of the Hindus were not only 

 greater than those of the Romans, but in certain directions sui- 





